Marzia Bisi

Grad’s equations and hydrodynamics for weakly inelastic granular flows

We introduce and discuss Grad’s moment equations for dilute granular systems of hard spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. An important by-product is that in this way we obtain the hydrodynamic description of a system of nearly elastic particles by a direct procedure from the Boltzmann equation, without resorting to any homogeneous cooling state assumption. Several crucial results of the pertinent literature are recovered in the present physical context in which deviation from elastic scattering is of the same order as the Knudsen number. In particular, the correlation function plays a fundamental role in the decay of the temperature, and the latter is described asymptotically, in space homogeneous conditions, by a corrected Haff’s law.

Shock wave structure for generalized Burnett equations

Stationary shock wave solutions for the generalized Burnett equations (GBE) [A. V. Bobylev, “Generalized Burnett hydrodynamics,” J. Stat. Phys. 132, 569 (2008)] are studied. Based on the results of Bisi et al. [“Qualitative analysis of the generalized Burnett equations and applications to half-space problems,” Kinet. Relat. Models 1, 295 (2008)], we choose a unique (optimal) form of GBE and solve numerically the shock wave problem for various Mach numbers. The results are compared with the numerical solutions of Navier–Stokes equations and with the Mott–Smith approximation for the Boltzmann equation (all calculations are done for Maxwell molecules) since it is believed that the Mott–Smith approximation yields better results for strong shocks. The comparison shows that GBE yield certain improvement of the Navier–Stokes results for moderate Mach numbers.

Fluid-dynamic equations for granular particles in a host medium

A kinetic model for a granular gas interacting with a given background by binary dissipative collisions is analyzed, with particular reference to the derivation of macroscopic equations for the fundamental observables. Particles are modelled as inelastic hard spheres under the assumption of collision dominated regime (small mean free path). Closure of the relevant moment equations is achieved by resorting to a maximum entropy principle, and two specific entropy functionals have been worked out in detail, in the class of the admissible ones for the relevant linear extended Boltzmann equation. Considered macroscopic fields include density, mass velocity, and granular temperature. In the hydrodynamic limit when the mean free path tends to zero, a single drift-diffusion equation of Navier-Stokes type is recovered for the only hydrodynamic variable of the physical problem.

Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions

Starting from a Boltzmann kinetic model for a gas mixture with bimolecular chemical reaction, hydrodynamic equations at Euler level are deduced by a consistent hydrodynamic limit in the presence of resonance, namely when the fast process driving evolution is constituted by elastic scattering between particles of the same species. The structure of the resulting multi-temperature and multi-velocity fluid-dynamic description is briefly commented on, and some results in closed analytical form are given for the special case of Maxwellian collision kernel.

Entropy dissipation estimates for the linear Boltzmann operator

Abstract We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignanis conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to [27] , [32] . For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.

Uniqueness in the Weakly Inelastic Regime of the Equilibrium State to the Boltzmann Equation Driven by a Particle Bath

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

\alpha \in (0,1)

Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: The steady shock problem

) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime, i.e., for any inelasticity parameter

Dilute Granular Flows in a Medium

\alpha \in (\alpha_0,1)

Kinetic Problems in Rarefied Gas Mixtures

, with some constructive